Working with Binary Numbers

Computers are, at the most fundamental level, just a collection of electrical switches. Numbers and characters are represented by the positions of these switches. Because a switch has only two positions , on or off, it uses a binary, or base 2, numbering system (the root bi means "two"). A base 2 system has just two digits: 0 and 1.

Computers usually group these digits into eight place values, known as a byte or an octet . The eight place values are:

2 ⁷ 2 ⁶ 2 ⁵ 2 ⁴ 2 ³ 2 ² 2 ¹ 2

The place values are calculated:

2 = 1

2 ¹ = 1 x 2 = 2

2 ² = 2 x 2 = 4

2 ³ = 4 x 2 = 8

2 ⁴ = 8 x 2 = 16

2 ⁵ = 16 x 2 = 32

2 ⁶ = 32 x 2 = 64

2 ⁷ = 64 x 2 = 128

So the place values of a binary octet are:

128 64 32 16 8 4 2 1

Thus the binary octet 10010111 can be read as follows :

1 x 128 = 128

0 x 64 = 0

0 x 32 = 0

1 x 16 = 16

0 x 8 = 0

1 x 4 = 4

1 x 2 = 2

1 x 1 = 1

or 128 + 16 + 4 + 2 + 1 = 151

Working in binary is easy because for every place value there is either one quantity of that value or none of that value. For another example, 11101001 = 128 + 64 + 32 + 8 + 1 = 233.

Where converting binary to decimal is a matter of adding the place values, converting from decimal to binary is a matter of subtracting place values. To convert the decimal number 178 to binary, for instance, begin by subtracting the highest base 2 place value possible from the number:

1. 178 is greater than 128, so we know there is a 1 at that place value: 178 “ 128 = 50.

2. 50 is less than 64, so there is a 0 at that place value.

3. 50 is greater than 32, so there is a 1 at that place value: 50 “ 32 = 18.

4. 18 is greater than 16, so there is a 1 at that place value: 18 “ 16 = 2.

5. 2 is less than 8, so there is a 0 at that place value.

6. 2 is less than 4, so there is a 0 at that place value.

7. 2 is equal to 2, so there is a 1 at that place value: 2 “ 2 = 0.

8. 0 is less than 1, so there is a 0 at that place value.

Putting the results of all these steps together, 178 is 10110010 in binary.

Another example may be helpful. Given 110:

1. 110 is less than 128, so there is a 0 at that place value.

2. 110 is greater than 64, so there is a 1 at that place value: 110 “ 64 = 46.

3. 46 is greater than 32, so there is a 1 at that place value: 46 “ 32 = 14.

4. 14 is less than 16, so there is a 0 at that place value.

5. 14 is greater than 8, so there is a 1 at that place value: 14 “ 8 = 6.

6. 6 is greater than 4, so there is a 1 at that place value: 6 “ 4 = 2.

7. There is a 1 at the 2 place value: 2 “ 2 = 0.

8. 0 is less than 1, so there is a 0 at that place value.

Therefore, 110 is 01101110 in binary.